A set $A \subseteq \mathbb {N}$ is a set of pointwise recurrence if for all minimal dynamical systems $(X, T)$, all $x \in X$, and all open neighborhoods $U \subseteq X$ of x, there exists a time $n \in A$ such that $T^n x \in U$. The set A is dynamically thick if the same holds for all non-empty, open sets $U \subseteq X$. Our main results give combinatorial characterizations of sets of pointwise recurrence and dynamically thick sets that allow us to answer questions of Host, Kra, and Maass [Variations on topological recurrence. Monatsh. Math. 179(1) (2016), 57–89] and Glasner, Tsankov, Weiss, and Zucker [Bernoulli disjointness. Duke Math. J. 170(4) (2021), 615–651]. We also introduce and study a local version of dynamical thickness called dynamical piecewise syndeticity. We show that dynamically piecewise syndetic sets are piecewise syndetic, generalizing results of Glasner and Weiss [Interpolation sets for subalgebras of $l^\infty(\mathbb{Z})$. Israel J. Math. 44(4) (1983), 345–360], Huang and Ye [Dynamical systems disjoint from any minimal system. Trans. Amer. Math. Soc. 357(2) (2005), 669–694], and Dong, Shao, and Ye [Product recurrent properties, disjointness and weak disjointness. Israel J. Math. 188 (2012), 463–507]. The proofs involve the algebra of families of large sets, dynamics on the space of ultrafilters, and our recent characterization of dynamically syndetic sets.